Chapter 4 Protect
Mike Aguilar
Constructing Portfolios
Note: Using Return.portfolio() does NOT include
transaction costs, nor does it produce an Optimal portfolio.
Buy&Hold Equal Weights
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)Buy&Hold NonEqual Weights
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
wF = .05
wGE = .25
wCAT = .70
startingweights<-c(wF,wGE,wCAT)
port1<-Return.portfolio(R=Returns, weights = startingweights, verbose = TRUE)Buy&Hold With Shorts
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
library(PerformanceAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
wF = -.05
wGE = .25
wCAT = .8
startingweights<-c(wF,wGE,wCAT)
port1<-Return.portfolio(R=Returns, weights = startingweights, verbose = TRUE)Buy&Hold Fully Invested
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
wF = .05
wGE = .25
wCAT = 1-wF-wGE
startingweights<-c(wF,wGE,wCAT)
port1<-Return.portfolio(R=Returns, weights = startingweights, verbose = TRUE)Buy&Hold Fully Invested v2
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
weightmatrix<-Returns #stealing the data type and dates
T = dim(weightmatrix)[1] #Number of days
wF = .05
wGE = .25
wCAT = .70
weightmatrix$F<-rep(wF,T)
weightmatrix$GE<-rep(wGE,T)
weightmatrix$CAT<-rep(wCAT,T)
weightmatrix$GE<-rep(.05,dim(weightmatrix$GE)[1]) # change a weight arbitrarily for this example
weightmatrix$CAT<-1-rowSums(weightmatrix[,-3]) # "-3" uses CAT to ensure the portfolio is always fully invested
head(weightmatrix)## F GE CAT
## 2015-01-05 0.05 0.05 0.9
## 2015-01-06 0.05 0.05 0.9
## 2015-01-07 0.05 0.05 0.9
## 2015-01-08 0.05 0.05 0.9
## 2015-01-09 0.05 0.05 0.9
## 2015-01-12 0.05 0.05 0.9port1<-Return.portfolio(R=Returns, weights = weightmatrix, verbose = TRUE)Compute Portfolio Value
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)
StartingEquity = 1 # change to whatever value you like
port1$GrossReturns<-port1$returns+1
port1$GrossReturns[1,1]=StartingEquity
colnames(port1$GrossReturns)<- "GrossReturns"
port1$PortfolioIndex=cumprod(port1$GrossReturns)
colnames(port1$PortfolioIndex)<- "PortfolioIndex"
tail(port1$PortfolioIndex)## PortfolioIndex
## 2021-04-22 1.632678
## 2021-04-23 1.653915
## 2021-04-26 1.657846
## 2021-04-27 1.662295
## 2021-04-28 1.664662
## 2021-04-29 1.606665chart.TimeSeries(port1$PortfolioIndex)
Compute Portfolio Value v2
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)
chart.CumReturns(port1$returns)
Portfolio Drawdown
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
port1<-Return.portfolio(R=Returns, verbose = TRUE)
chart.Drawdown(port1$returns)
Portfolio Value/Return Drawdown
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
port1<-Return.portfolio(R=Returns, verbose = TRUE)
charts.PerformanceSummary(port1$returns)
Chart Evolution of Weights
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)
chart.StackedBar(port1$BOP.Weight["/2015-02-28",])
Pie Chart of Weights
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)
LastWeight = data.matrix(tail(port1$EOP.Weight,n=1))
pie(LastWeight,Tickers)
Chart Asset Contributions
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
port1<-Return.portfolio(R=Returns, verbose = TRUE)
chart.TimeSeries(port1$contribution,legend.loc = 'bottomright')
Contributions<-cumsum(port1$contribution)
chart.TimeSeries(Contributions,legend.loc = 'topleft')
Regular Rebalancing
Note: Using Return.portfolio does NOT include transaction costs, nor does it produce an Optimal portfolio. Can alter rebalance_on to (days,weeks,months,quarters,years).
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
library(PerformanceAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
wF = .05
wGE = .25
wCAT = .70
startingweights<-c(wF,wGE,wCAT)
port1<-Return.portfolio(R=Returns, weights = startingweights, rebalance_on = "days", verbose = TRUE)
head(port1$BOP.Weight)## daily.returns daily.returns.1 daily.returns.2
## 2015-01-05 0.05 0.25 0.7
## 2015-01-06 0.05 0.25 0.7
## 2015-01-07 0.05 0.25 0.7
## 2015-01-08 0.05 0.25 0.7
## 2015-01-09 0.05 0.25 0.7
## 2015-01-12 0.05 0.25 0.7head(port1$EOP.Weight)## daily.returns daily.returns.1 daily.returns.2
## 2015-01-05 0.05023157 0.2565698 0.6931986
## 2015-01-06 0.05004443 0.2471757 0.7027799
## 2015-01-07 0.05080701 0.2470435 0.7021494
## 2015-01-08 0.05068304 0.2501469 0.6991701
## 2015-01-09 0.04994504 0.2496409 0.7004140
## 2015-01-12 0.05036309 0.2511264 0.6985105Irregular Rebalancing
You can control the precise date of rebalancing by defining a weight matrix that contains the weights for each period on every ticker.
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PortfolioAnalytics)
Tickers<-c('F','GE','CAT')
startd = "2015-01-01"
endd = "2021-04-30"
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "F" "GE" "CAT"Prices = do.call(merge,lapply(Tickers, function(x) Ad(get(x))))
Prices = na.omit(Prices[-1,])
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-names(Prices)
weightmatrix<-Returns #stealing the data type and dates
T = dim(weightmatrix)[1] #Number of days
wF = .05
wGE = .25
wCAT = .70
weightmatrix$F.Adjusted<-rep(wF,T)
weightmatrix$GE.Adjusted<-rep(wGE,T)
weightmatrix$CAT.Adjusted<-rep(wCAT,T)
port1<-Return.portfolio(R=Returns, weights = weightmatrix, verbose = TRUE)
head(port1$BOP.Weight)## F.Adjusted GE.Adjusted CAT.Adjusted
## 2015-01-06 0.05 0.25 0.7
## 2015-01-07 0.05 0.25 0.7
## 2015-01-08 0.05 0.25 0.7
## 2015-01-09 0.05 0.25 0.7
## 2015-01-12 0.05 0.25 0.7
## 2015-01-13 0.05 0.25 0.7Optimal Portfolios
Min Variance-Long Only
Minimum Variance, Subject to Target Return, Fully Invested, Long Only
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a target return constraint
init.portf <- add.constraint(portfolio=init.portf, type="return", return_target=0.001)
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add a long only constraint such that the weight of an asset is between 0 and 1
init.portf <- add.constraint(portfolio=init.portf, type="long_only")
# Add objective to minimize portfolio variance
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "var")
### Solve Portfolio: Minimize Variance
library(ROI)
opt.MinVar <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
trace=TRUE)
summary(opt.MinVar) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.1718 0.0000 0.0972 0.7310
##
## Objective Measures:
## StdDev
## 0.01529
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - return
## - full_investment
## - long_only
##
## Objectives:
## Enabled objective names
## - var
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## IBM GE F MSFT
## 0 0 0 0
## max:
## IBM GE F MSFT
## 1 1 1 1
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 3
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 0
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.426643
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.2405153
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: portfolio_risk_objective
## $name
## [1] "var"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "var")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.03888774 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MinVar)
sum(OptWeight)## [1] 1chart.Weights(opt.MinVar)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.001# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0002338935Min Variance-Long/Short
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a target return constraint
init.portf <- add.constraint(portfolio=init.portf, type="return", return_target=0.001)
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add objective to minimize portfolio variance
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "var")
### Solve Portfolio: Minimize Variance
library(ROI)
opt.MinVar <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
trace=TRUE)
summary(opt.MinVar) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.1923 -0.0286 0.1114 0.7249
##
## Objective Measures:
## StdDev
## 0.01528
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - return
## - full_investment
##
## Objectives:
## Enabled objective names
## - var
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.4243172
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.2374474
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: portfolio_risk_objective
## $name
## [1] "var"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "var")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.009923935 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MinVar)
sum(OptWeight)## [1] 1chart.Weights(opt.MinVar)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.001# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0002335735Max Sharpe-LongOnly
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Solve Portfolio
library(ROI)
opt <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
summary(opt) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.0218 -0.0804 0.0757 0.9830
##
## Objective Measures:
## StdDev
## 0.01733
##
##
## mean
## 0.001288
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.02113154
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.3664756
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.2770972 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt)
sum(OptWeight)## [1] 1chart.Weights(opt)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.001287984# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0003003911Max Sharpe-Long/Short
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Solve Portfolio
library(ROI)
opt <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
summary(opt) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.0218 -0.0804 0.0757 0.9830
##
## Objective Measures:
## StdDev
## 0.01733
##
##
## mean
## 0.001288
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.02113156
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.3664756
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.2745883 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt)
sum(OptWeight)## [1] 1chart.Weights(opt)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.001287984# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0003003909Max Sharpe-Weight Bounds
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add upper and lower bounds on the weights of assets
init.portf <- add.constraint(portfolio=init.portf, type="box", min=0, max=0.5)
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Solve Portfolio
library(ROI)
opt <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
summary(opt) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.3541 0.0000 0.1459 0.5000
##
## Objective Measures:
## StdDev
## 0.01426
##
##
## mean
## 0.0007529
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
## - box
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## IBM GE F MSFT
## 0 0 0 0
## max:
## IBM GE F MSFT
## 0.5 0.5 0.5 0.5
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 3
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 0
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.6033234
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.1770536
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.202183 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt)
sum(OptWeight)## [1] 1chart.Weights(opt)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.0007528538# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0002033752Max Sharpe-Limit #Positions
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Returns))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add position limit constraint such that we have a maximum number of long positions and short positions
init.portf <- add.constraint(portfolio=init.portf,
type="position_limit",
max_pos_long = 3,
max_pos_short = 2)
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Solve Portfolio
library(ROI)
opt <- optimize.portfolio(R=Returns,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
summary(opt) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Returns, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.0218 -0.0804 0.0757 0.9830
##
## Objective Measures:
## StdDev
## 0.01733
##
##
## mean
## 0.001288
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Returns))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
## - position_limit
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] 3
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] 2
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.02113154
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.3664756
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.2668672 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt)
sum(OptWeight)## [1] 1chart.Weights(opt)
# Portfolio returns
PortReturn <- Return.portfolio(R = Returns,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
mean(PortReturn)## [1] 0.001287984# Variance
var(PortReturn)## portfolio.returns
## portfolio.returns 0.0003003908Max Sharpe-FactorExposure
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Return = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Return = na.omit(Return[-1,])
colnames(Return)<-Tickers
### Estimate Market Risk Exposure(assume Rf = 0)
# Load market data(we need this because we want it to calculate the market risk exposure)
getSymbols("^GSPC", from = startd, to = endd)## [1] "^GSPC"SP500 <- periodReturn(Ad(get("GSPC")),period='daily',type='arithmetic')
na.omit(SP500)## daily.returns
## 2015-01-02 0.000000e+00
## 2015-01-05 -1.827811e-02
## 2015-01-06 -8.893472e-03
## 2015-01-07 1.162984e-02
## 2015-01-08 1.788828e-02
## 2015-01-09 -8.403811e-03
## 2015-01-12 -8.093685e-03
## 2015-01-13 -2.578555e-03
## 2015-01-14 -5.813067e-03
## 2015-01-15 -9.247876e-03
## 2015-01-16 1.342420e-02
## 2015-01-20 1.549952e-03
## 2015-01-21 4.731624e-03
## 2015-01-22 1.526972e-02
## 2015-01-23 -5.491522e-03
## 2015-01-26 2.568461e-03
## 2015-01-27 -1.338786e-02
## 2015-01-28 -1.349561e-02
## 2015-01-29 9.534685e-03
## 2015-01-30 -1.299197e-02
## 2015-02-02 1.296246e-02
## 2015-02-03 1.443949e-02
## 2015-02-04 -4.156046e-03
## 2015-02-05 1.029141e-02
## 2015-02-06 -3.418172e-03
## 2015-02-09 -4.247195e-03
## 2015-02-10 1.067556e-02
## 2015-02-11 -2.903379e-05
## 2015-02-12 9.644506e-03
## 2015-02-13 4.074739e-03
## 2015-02-17 1.597575e-03
## 2015-02-18 -3.143091e-04
## 2015-02-19 -1.062058e-03
## 2015-02-20 6.126534e-03
## 2015-02-23 -3.033393e-04
## 2015-02-24 2.758771e-03
## 2015-02-25 -7.657236e-04
## 2015-02-26 -1.476028e-03
## 2015-02-27 -2.956304e-03
## 2015-03-02 6.124919e-03
## 2015-03-03 -4.538542e-03
## 2015-03-04 -4.388503e-03
## 2015-03-05 1.196080e-03
## 2015-03-06 -1.417395e-02
## 2015-03-09 3.944421e-03
## 2015-03-10 -1.696133e-02
## 2015-03-11 -1.917680e-03
## 2015-03-12 1.260144e-02
## 2015-03-13 -6.074711e-03
## 2015-03-16 1.353367e-02
## 2015-03-17 -3.320174e-03
## 2015-03-18 1.215842e-02
## 2015-03-19 -4.872579e-03
## 2015-03-20 9.012755e-03
## 2015-03-23 -1.745731e-03
## 2015-03-24 -6.139422e-03
## 2015-03-25 -1.455891e-02
## 2015-03-26 -2.377500e-03
## 2015-03-27 2.368562e-03
## 2015-03-30 1.223664e-02
## 2015-03-31 -8.795775e-03
## 2015-04-01 -3.965372e-03
## 2015-04-02 3.529667e-03
## 2015-04-06 6.608815e-03
## 2015-04-07 -2.061904e-03
## 2015-04-08 2.682533e-03
## 2015-04-09 4.457481e-03
## 2015-04-10 5.202865e-03
## 2015-04-13 -4.581281e-03
## 2015-04-14 1.629759e-03
## 2015-04-15 5.148196e-03
## 2015-04-16 -7.784438e-04
## 2015-04-17 -1.131125e-02
## 2015-04-20 9.235131e-03
## 2015-04-21 -1.480605e-03
## 2015-04-22 5.087480e-03
## 2015-04-23 2.357716e-03
## 2015-04-24 2.252800e-03
## 2015-04-27 -4.141314e-03
## 2015-04-28 2.769232e-03
## 2015-04-29 -3.740336e-03
## 2015-04-30 -1.012891e-02
## 2015-05-01 1.092300e-02
## 2015-05-04 2.940749e-03
## 2015-05-05 -1.183738e-02
## 2015-05-06 -4.455725e-03
## 2015-05-07 3.773814e-03
## 2015-05-08 1.345790e-02
## 2015-05-11 -5.089561e-03
## 2015-05-12 -2.949638e-03
## 2015-05-13 -3.049549e-04
## 2015-05-14 1.077929e-02
## 2015-05-15 7.684135e-04
## 2015-05-18 3.047948e-03
## 2015-05-19 -6.433745e-04
## 2015-05-20 -9.305160e-04
## 2015-05-21 2.337874e-03
## 2015-05-22 -2.233886e-03
## 2015-05-26 -1.028198e-02
## 2015-05-27 9.162641e-03
## 2015-05-28 -1.266761e-03
## 2015-05-29 -6.318469e-03
## 2015-06-01 2.059461e-03
## 2015-06-02 -1.008596e-03
## 2015-06-03 2.118871e-03
## 2015-06-04 -8.623167e-03
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## 2019-06-18 9.717400e-03
## 2019-06-19 2.985164e-03
## 2019-06-20 9.472185e-03
## 2019-06-21 -1.259223e-03
## 2019-06-24 -1.731887e-03
## 2019-06-25 -9.496397e-03
## 2019-06-26 -1.233934e-03
## 2019-06-27 3.823176e-03
## 2019-06-28 5.757453e-03
## 2019-07-01 7.672301e-03
## 2019-07-02 2.928126e-03
## 2019-07-03 7.672378e-03
## 2019-07-05 -1.805902e-03
## 2019-07-08 -4.835444e-03
## 2019-07-09 1.236557e-03
## 2019-07-10 4.510689e-03
## 2019-07-11 2.285227e-03
## 2019-07-12 4.620175e-03
## 2019-07-15 1.758691e-04
## 2019-07-16 -3.403779e-03
## 2019-07-17 -6.531244e-03
## 2019-07-18 3.581998e-03
## 2019-07-19 -6.176735e-03
## 2019-07-22 2.828695e-03
## 2019-07-23 6.847483e-03
## 2019-07-24 4.688148e-03
## 2019-07-25 -5.262401e-03
## 2019-07-26 7.387691e-03
## 2019-07-29 -1.616114e-03
## 2019-07-30 -2.578655e-03
## 2019-07-31 -1.088553e-02
## 2019-08-01 -8.998794e-03
## 2019-08-02 -7.282740e-03
## 2019-08-05 -2.977782e-02
## 2019-08-06 1.301702e-02
## 2019-08-07 7.668759e-04
## 2019-08-08 1.876230e-02
## 2019-08-09 -6.616607e-03
## 2019-08-12 -1.231732e-02
## 2019-08-13 1.513169e-02
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## 2019-08-15 2.464268e-03
## 2019-08-16 1.442612e-02
## 2019-08-19 1.210587e-02
## 2019-08-20 -7.914727e-03
## 2019-08-21 8.246799e-03
## 2019-08-22 -5.060750e-04
## 2019-08-23 -2.594634e-02
## 2019-08-26 1.098299e-02
## 2019-08-27 -3.203181e-03
## 2019-08-28 6.545480e-03
## 2019-08-29 1.268729e-02
## 2019-08-30 6.427873e-04
## 2019-09-03 -6.899100e-03
## 2019-09-04 1.084208e-02
## 2019-09-05 1.300981e-02
## 2019-09-06 9.106052e-04
## 2019-09-09 -9.401016e-05
## 2019-09-10 3.223044e-04
## 2019-09-11 7.229681e-03
## 2019-09-12 2.879153e-03
## 2019-09-13 -7.244141e-04
## 2019-09-16 -3.135587e-03
## 2019-09-17 2.581752e-03
## 2019-09-18 3.426919e-04
## 2019-09-19 1.997486e-05
## 2019-09-20 -4.895577e-03
## 2019-09-23 -9.693590e-05
## 2019-09-24 -8.416371e-03
## 2019-09-25 6.158572e-03
## 2019-09-26 -2.428916e-03
## 2019-09-27 -5.316352e-03
## 2019-09-30 5.047607e-03
## 2019-10-01 -1.225837e-02
## 2019-10-02 -1.790320e-02
## 2019-10-03 7.971913e-03
## 2019-10-04 1.421690e-02
## 2019-10-07 -4.478295e-03
## 2019-10-08 -1.556082e-02
## 2019-10-09 9.104492e-03
## 2019-10-10 6.415696e-03
## 2019-10-11 1.093898e-02
## 2019-10-14 -1.387119e-03
## 2019-10-15 9.955677e-03
## 2019-10-16 -1.999543e-03
## 2019-10-17 2.762832e-03
## 2019-10-18 -3.919345e-03
## 2019-10-21 6.871616e-03
## 2019-10-22 -3.568667e-03
## 2019-10-23 2.847149e-03
## 2019-10-24 1.920446e-03
## 2019-10-25 4.072701e-03
## 2019-10-28 5.581338e-03
## 2019-10-29 -8.324052e-04
## 2019-10-30 3.253370e-03
## 2019-10-31 -3.022861e-03
## 2019-11-01 9.662312e-03
## 2019-11-04 3.704089e-03
## 2019-11-05 -1.185699e-03
## 2019-11-06 7.024972e-04
## 2019-11-07 2.730095e-03
## 2019-11-08 2.560676e-03
## 2019-11-11 -1.962467e-03
## 2019-11-12 1.564646e-03
## 2019-11-13 7.115345e-04
## 2019-11-14 8.370428e-04
## 2019-11-15 7.695488e-03
## 2019-11-18 5.031527e-04
## 2019-11-19 -5.925942e-04
## 2019-11-20 -3.756184e-03
## 2019-11-21 -1.582752e-03
## 2019-11-22 2.174936e-03
## 2019-11-25 7.507292e-03
## 2019-11-26 2.195570e-03
## 2019-11-27 4.174424e-03
## 2019-11-29 -4.011220e-03
## 2019-12-02 -8.631021e-03
## 2019-12-03 -6.638095e-03
## 2019-12-04 6.323568e-03
## 2019-12-05 1.500251e-03
## 2019-12-06 9.135724e-03
## 2019-12-09 -3.162821e-03
## 2019-12-10 -1.096934e-03
## 2019-12-11 2.908158e-03
## 2019-12-12 8.575226e-03
## 2019-12-13 7.258195e-05
## 2019-12-16 7.147785e-03
## 2019-12-17 3.352924e-04
## 2019-12-18 -4.323002e-04
## 2019-12-19 4.459292e-03
## 2019-12-20 4.944781e-03
## 2019-12-23 8.661436e-04
## 2019-12-24 -1.954482e-04
## 2019-12-26 5.128167e-03
## 2019-12-27 3.398490e-05
## 2019-12-30 -5.780823e-03
## 2019-12-31 2.946022e-03
## 2020-01-02 8.378803e-03
## 2020-01-03 -7.059871e-03
## 2020-01-06 3.533373e-03
## 2020-01-07 -2.803238e-03
## 2020-01-08 4.902451e-03
## 2020-01-09 6.655262e-03
## 2020-01-10 -2.855179e-03
## 2020-01-13 6.976215e-03
## 2020-01-14 -1.514533e-03
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## 2020-01-16 8.366553e-03
## 2020-01-17 3.862162e-03
## 2020-01-21 -2.651978e-03
## 2020-01-22 2.890761e-04
## 2020-01-23 1.140977e-03
## 2020-01-24 -9.042161e-03
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## 2020-01-28 1.005358e-02
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## 2020-02-03 7.254614e-03
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## 2020-05-15 3.926363e-03
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## 2020-06-18 5.942200e-04
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## 2020-08-05 6.429743e-03
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## 2020-08-17 2.709842e-03
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## 2020-08-19 -4.404395e-03
## 2020-08-20 3.158633e-03
## 2020-08-21 3.441107e-03
## 2020-08-24 1.004372e-02
## 2020-08-25 3.596351e-03
## 2020-08-26 1.019563e-02
## 2020-08-27 1.673044e-03
## 2020-08-28 6.732565e-03
## 2020-08-31 -2.194963e-03
## 2020-09-01 7.525003e-03
## 2020-09-02 1.536591e-02
## 2020-09-03 -3.512584e-02
## 2020-09-04 -8.133027e-03
## 2020-09-08 -2.775634e-02
## 2020-09-09 2.014499e-02
## 2020-09-10 -1.758480e-02
## 2020-09-11 5.330724e-04
## 2020-09-14 1.274183e-02
## 2020-09-15 5.219360e-03
## 2020-09-16 -4.618947e-03
## 2020-09-17 -8.412366e-03
## 2020-09-18 -1.118258e-02
## 2020-09-21 -1.157110e-02
## 2020-09-22 1.051794e-02
## 2020-09-23 -2.372145e-02
## 2020-09-24 2.987459e-03
## 2020-09-25 1.597672e-02
## 2020-09-28 1.611059e-02
## 2020-09-29 -4.812665e-03
## 2020-09-30 8.253718e-03
## 2020-10-01 5.292908e-03
## 2020-10-02 -9.577652e-03
## 2020-10-05 1.797271e-02
## 2020-10-06 -1.397352e-02
## 2020-10-07 1.739675e-02
## 2020-10-08 8.010124e-03
## 2020-10-09 8.793533e-03
## 2020-10-12 1.641581e-02
## 2020-10-13 -6.306919e-03
## 2020-10-14 -6.623142e-03
## 2020-10-15 -1.527755e-03
## 2020-10-16 1.349196e-04
## 2020-10-19 -1.632986e-02
## 2020-10-20 4.727334e-03
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## 2020-10-22 5.218925e-03
## 2020-10-23 3.445762e-03
## 2020-10-26 -1.858952e-02
## 2020-10-27 -3.025619e-03
## 2020-10-28 -3.528788e-02
## 2020-10-29 1.194733e-02
## 2020-10-30 -1.212955e-02
## 2020-11-02 1.231820e-02
## 2020-11-03 1.779929e-02
## 2020-11-04 2.204705e-02
## 2020-11-05 1.946019e-02
## 2020-11-06 -2.877153e-04
## 2020-11-09 1.169989e-02
## 2020-11-10 -1.399795e-03
## 2020-11-11 7.651855e-03
## 2020-11-12 -9.978532e-03
## 2020-11-13 1.361034e-02
## 2020-11-16 1.164805e-02
## 2020-11-17 -4.791926e-03
## 2020-11-18 -1.156383e-02
## 2020-11-19 3.946442e-03
## 2020-11-20 -6.792563e-03
## 2020-11-23 5.635931e-03
## 2020-11-24 1.616167e-02
## 2020-11-25 -1.584418e-03
## 2020-11-27 2.396979e-03
## 2020-11-30 -4.595549e-03
## 2020-12-01 1.127119e-02
## 2020-12-02 1.791167e-03
## 2020-12-03 -6.241572e-04
## 2020-12-04 8.836275e-03
## 2020-12-07 -1.935638e-03
## 2020-12-08 2.787148e-03
## 2020-12-09 -7.949202e-03
## 2020-12-10 -1.285108e-03
## 2020-12-11 -1.264997e-03
## 2020-12-14 -4.359259e-03
## 2020-12-15 1.292125e-02
## 2020-12-16 1.772795e-03
## 2020-12-17 5.757655e-03
## 2020-12-18 -3.511118e-03
## 2020-12-21 -3.906279e-03
## 2020-12-22 -2.073093e-03
## 2020-12-23 7.458113e-04
## 2020-12-24 3.536589e-03
## 2020-12-28 8.722529e-03
## 2020-12-29 -2.227380e-03
## 2020-12-30 1.341547e-03
## 2020-12-31 6.438845e-03
## 2021-01-04 -1.475483e-02
## 2021-01-05 7.082595e-03
## 2021-01-06 5.709843e-03
## 2021-01-07 1.484740e-02
## 2021-01-08 5.491863e-03
## 2021-01-11 -6.554751e-03
## 2021-01-12 4.157885e-04
## 2021-01-13 2.275642e-03
## 2021-01-14 -3.753451e-03
## 2021-01-15 -7.190028e-03
## 2021-01-19 8.136379e-03
## 2021-01-20 1.393563e-02
## 2021-01-21 3.167231e-04
## 2021-01-22 -3.010611e-03
## 2021-01-25 3.615839e-03
## 2021-01-26 -1.488834e-03
## 2021-01-27 -2.567788e-02
## 2021-01-28 9.760626e-03
## 2021-01-29 -1.931148e-02
## 2021-02-01 1.605177e-02
## 2021-02-02 1.389822e-02
## 2021-02-03 1.008769e-03
## 2021-02-04 1.085332e-02
## 2021-02-05 3.897495e-03
## 2021-02-08 7.399348e-03
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## 2021-02-10 -3.451848e-04
## 2021-02-11 1.662455e-03
## 2021-02-12 4.711033e-03
## 2021-02-16 -5.692724e-04
## 2021-02-17 -3.204021e-04
## 2021-02-18 -4.415836e-03
## 2021-02-19 -1.854897e-03
## 2021-02-22 -7.732839e-03
## 2021-02-23 1.256318e-03
## 2021-02-24 1.135161e-02
## 2021-02-25 -2.447881e-02
## 2021-02-26 -4.750214e-03
## 2021-03-01 2.379076e-02
## 2021-03-02 -8.080852e-03
## 2021-03-03 -1.306622e-02
## 2021-03-04 -1.341721e-02
## 2021-03-05 1.949597e-02
## 2021-03-08 -5.359231e-03
## 2021-03-09 1.415464e-02
## 2021-03-10 6.030314e-03
## 2021-03-11 1.039549e-02
## 2021-03-12 1.015398e-03
## 2021-03-15 6.491921e-03
## 2021-03-16 -1.569684e-03
## 2021-03-17 2.879382e-03
## 2021-03-18 -1.476054e-02
## 2021-03-19 -6.027039e-04
## 2021-03-22 7.025118e-03
## 2021-03-23 -7.630854e-03
## 2021-03-24 -5.467336e-03
## 2021-03-25 5.240266e-03
## 2021-03-26 1.663120e-02
## 2021-03-29 -8.680126e-04
## 2021-03-30 -3.157833e-03
## 2021-03-31 3.622499e-03
## 2021-04-01 1.182520e-02
## 2021-04-05 1.443823e-02
## 2021-04-06 -9.735308e-04
## 2021-04-07 1.475233e-03
## 2021-04-08 4.220633e-03
## 2021-04-09 7.719934e-03
## 2021-04-12 -1.960790e-04
## 2021-04-13 3.294487e-03
## 2021-04-14 -4.087727e-03
## 2021-04-15 1.109419e-02
## 2021-04-16 3.608820e-03
## 2021-04-19 -5.306560e-03
## 2021-04-20 -6.802320e-03
## 2021-04-21 9.306056e-03
## 2021-04-22 -9.210658e-03
## 2021-04-23 1.092870e-02
## 2021-04-26 1.782271e-03
## 2021-04-27 -2.148958e-04
## 2021-04-28 -8.455399e-04
## 2021-04-29 6.762807e-03SP500 <- SP500[-1,] # remove the first row
names(SP500) <- "SP500"
Betas <- sapply(colnames(Return), function(x) lm(Return[,x]~1+SP500)$coef[2]) #First regression
MktExposure <- lm(colMeans(Return)~1+Betas)$coef[2] #Second regression### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Return))
# Add a factor exposure constraint: the factor exposure constraint allows the user to set upper and lower bounds on exposures to risk factors. The exposures can be passed in as a vector or matrix. Here we specify a vector for B with arbitrary values, e.g. betas of the assets, with a market risk exposure range from "lower" to "upper" of the estimated value
init.portf <- add.constraint(portfolio=init.portf,
type="factor_exposure",
B = scale(Betas),
lower=MktExposure*0.2,
upper=MktExposure*0.8)
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")### Solve Portfolio: Maximize Sharpe Ratio
# The default action if “mean” and “StdDev” are specified as objectives with optimize_method="ROI" is to maximize quadratic utility. If we want to maximize Sharpe Ratio, we need to pass in maxSR=TRUE to optimize.portfolio
library(ROI)
opt.MaxSR <- optimize.portfolio(R=Return,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
# Warning: Inf or -Inf values detected in box constraints, maximum return objectives must have finite box constraint values.
summary(opt.MaxSR) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Return, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.4256 -0.0625 0.1191 0.5178
##
## Objective Measures:
## StdDev
## 0.01446
##
##
## mean
## 0.0007866
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Return))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - factor_exposure
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.5326631
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.2216893
##
## Factor Exposure Constraints:
## Factor Exposure B Matrix:
## factor1
## IBM.SP500 -1.2658780
## GE.SP500 0.3171101
## F.SP500 -0.1763517
## MSFT.SP500 1.1251196
## attr(,"scaled:center")
## [1] 1.09136
## attr(,"scaled:scale")
## [1] 0.09927187
##
## Lower bound on factor exposures, lower:
## factor1
## 0.0007585556
##
## Upper bound on group weights, upper:
## factor1
## 0.003034222
##
## Realized Factor Exposures:
## factor1
## 0.003034222
##
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.274622 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MaxSR)
chart.Weights(opt.MaxSR)
# Sharpe ratio
SharpeRatio(Return, weights = OptWeight, FUN = "StdDev")## [,1]
## StdDev Sharpe (Rf=0%, p=95%): 0.05439822# Portfolio returns
PortReturn <- Return.portfolio(R = Return,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
Max Sharpe-TransactionCost
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Return = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Return = na.omit(Return[-1,])
colnames(Return)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Return))
# Relax the leverage constraint(min_sum = max_sum = 1, by default)
init.portf <- add.constraint(portfolio=init.portf, type="leverage",
min_sum = 0.99, max_sum = 1.01)
# Simplify the optimization problem by setting long only constraint
init.portf <- add.constraint(portfolio=init.portf, type="long_only")
# Add a transaction cost constraint: the transaction cost constraint allows the user to specify proportional transaction costs. Transaction costs are supported as a penalty for the global numeric solvers. Here we add the transaction cost constraint with the proportional transaction cost value of 0.1%
init.portf <- add.constraint(portfolio=init.portf, type="transaction_cost", ptc=0.001)
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Solve Portfolio: Maximize Sharpe Ratio
# The default action if “mean” and “StdDev” are specified as objectives with optimize_method="ROI" is to maximize quadratic utility.
# Note that with the ROI solvers, proportional transaction cost constraint is currently only supported for the global minimum variance and quadratic utility problems with ROI quadprog plugin
library(DEoptim)
opt.MaxSR <- optimize.portfolio(R=Return,
portfolio=init.portf,
maxSR=TRUE,
trace=TRUE)## Iteration: 1 bestvalit: 0.014043 bestmemit: 0.483926 0.030000 0.200000 0.282000
## Iteration: 2 bestvalit: 0.014004 bestmemit: 0.360000 0.134000 0.154000 0.352000
## Iteration: 3 bestvalit: 0.013997 bestmemit: 0.276000 0.098000 0.172000 0.454000
## Iteration: 4 bestvalit: 0.013997 bestmemit: 0.276000 0.098000 0.172000 0.454000
## Iteration: 5 bestvalit: 0.013905 bestmemit: 0.289364 0.163561 0.186120 0.352000
## Iteration: 6 bestvalit: 0.013905 bestmemit: 0.289364 0.163561 0.186120 0.352000
## Iteration: 7 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 8 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 9 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 10 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 11 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 12 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 13 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## Iteration: 14 bestvalit: 0.013863 bestmemit: 0.358816 0.094702 0.137990 0.398612
## [1] 0.35881624 0.09470237 0.13799032 0.39861197summary(opt.MaxSR) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Return, portfolio = init.portf, trace = TRUE,
## maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.3588 0.0947 0.1380 0.3986
##
## Objective Measures:
## mean
## 0.0006204
##
##
## StdDev
## 0.01396
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Return))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - leverage
## - long_only
## - transaction_cost
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 0.99
## max_sum = 1.01
## actual_leverage = 0.9901209
##
## Box Constraints:
## min:
## IBM GE F MSFT
## 0 0 0 0
## max:
## IBM GE F MSFT
## 1 1 1 1
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 4
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 0
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.6843495
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.1311839
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 2.183314 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MaxSR)
chart.Weights(opt.MaxSR)
# Sharpe ratio
SharpeRatio(Return, weights = OptWeight, FUN = "StdDev")## [,1]
## StdDev Sharpe (Rf=0%, p=95%): 0.04444737# Portfolio returns
PortReturn <- Return.portfolio(R = Return,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
Max Sharpe-Reoptimizing
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Return = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Return = na.omit(Return[-1,])
colnames(Return)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Return))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add risk and return objective(we need them to calculate the Sharpe ratio)
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")### Solve Portfolio: Maximize Sharpe Ratio
# The default action if “mean” and “StdDev” are specified as objectives with optimize_method="ROI" is to maximize quadratic utility. If we want to maximize Sharpe Ratio, we need to pass in maxSR=TRUE to optimize.portfolio
library(ROI)
opt.MaxSR <- optimize.portfolio(R=Return,
portfolio=init.portf,
optimize_method="ROI",
maxSR=TRUE,
trace=TRUE)
# Warning: Inf or -Inf values detected in box constraints, maximum return objectives must have finite box constraint values.
summary(opt.MaxSR) # The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Return, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE, maxSR = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## 0.0218 -0.0804 0.0757 0.9830
##
## Objective Measures:
## StdDev
## 0.01733
##
##
## mean
## 0.001288
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Return))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
##
## Objectives:
## Enabled objective names
## - mean
## - StdDev
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 3
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 1
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] 0.02113157
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 0.3664757
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "StdDev"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "StdDev")
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.3548465 secsCaution: The function is called “rebalancing” but the implementation reoptimizes the weights.
### Solve Portfolio: Maximize Sharpe Ratio with Re-balancing
# Quarters rebalancing with 4 year training period and 1 year rolling window
opt.MaxSR.Rebal <- optimize.portfolio.rebalancing(
R = Return,
portfolio = init.portf,
optimize_method = 'ROI',
maxSR=TRUE,
trace = TRUE,
rebalance_on = 'quarters',
training_period = 4*252,
rolling_window = 252)
# Warning: Inf or -Inf values detected in box constraints, maximum return objectives must have finite box constraint values.
summary(opt.MaxSR.Rebal) ## **************************************************
## PortfolioAnalytics Optimization with Rebalancing
## **************************************************
##
## Call:
## optimize.portfolio.rebalancing(R = Return, portfolio = init.portf,
## optimize_method = "ROI", trace = TRUE, maxSR = TRUE, rebalance_on = "quarters",
## training_period = 4 * 252, rolling_window = 252)
##
## First rebalance date:
## [1] "2019-03-29"
##
## Last rebalance date:
## [1] "2021-04-29"
##
## Annualized Portfolio Rebalancing Return:
## [1] 0.3164928
##
## Annualized Portfolio Standard Deviation:
## [1] 0.3560959
##
## Downside Risk Measures:
## portfolio.returns
## Semi Deviation 0.0162
## Gain Deviation 0.0160
## Loss Deviation 0.0178
## Downside Deviation (MAR=210%) 0.0198
## Downside Deviation (Rf=0%) 0.0156
## Downside Deviation (0%) 0.0156
## Maximum Drawdown 0.3053
## Historical VaR (95%) -0.0312
## Historical ES (95%) -0.0519
## Modified VaR (95%) -0.0326
## Modified ES (95%) -0.0546### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MaxSR)
OptWeight.Rebal = extractWeights(opt.MaxSR.Rebal)
chart.Weights(opt.MaxSR)
chart.Weights(opt.MaxSR.Rebal)
# Portfolio return
PortReturn <- Return.portfolio(R = Return,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
PortReturn.Rebal <- Return.portfolio(R = Return,
weights = OptWeight.Rebal,
geometric = FALSE)
Portfolio.R = cbind(PortReturn,PortReturn.Rebal)
names(Portfolio.R) = c("Non-Rebalancing","Rebalancing")
# First rebalance date: 2019-03-29
charts.PerformanceSummary(Portfolio.R, colors=c("darkred","steelblue"), main=" ") 
Max Utility
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Return = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Return = na.omit(Return[-1,])
colnames(Return)<-Tickers
### Construct Portfolio
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Return))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add risk and return objective(we need them to calculate the quadratic utility
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "var",risk_aversion=.25)
### Solve Portfolio: Maximize Quadratic Utility
# The default action if “mean” and “StdDev” are specified as objectives with optimize_method="ROI" is to maximize quadratic utility.
library(ROI)
opt.MaxQU <- optimize.portfolio(R=Return,
portfolio=init.portf,
optimize_method="ROI",
trace=TRUE)
# Warning: Inf or -Inf values detected in box constraints, maximum return objectives must have finite box constraint values.
summary(opt.MaxQU) #The optimization summary should be read carefully since not every constraint is guaranteed to be satisfied## **************************************************
## PortfolioAnalytics Optimization Summary
## **************************************************
##
## Call:
## optimize.portfolio(R = Return, portfolio = init.portf, optimize_method = "ROI",
## trace = TRUE)
##
## Optimal Weights:
## IBM GE F MSFT
## -5.3499 -1.7132 -1.0482 9.1113
##
## Objective Measures:
## mean
## 0.01036
##
##
## StdDev
## 0.1406
##
##
## Portfolio Assets and Initial Weights:
## IBM GE F MSFT
## 0.25 0.25 0.25 0.25
##
## **************************************************
## PortfolioAnalytics Portfolio Specification
## **************************************************
##
## Call:
## portfolio.spec(assets = colnames(Return))
##
## Number of assets: 4
## Asset Names
## [1] "IBM" "GE" "F" "MSFT"
##
## Constraints
## Enabled constraint types
## - full_investment
##
## Objectives:
## Enabled objective names
## - mean
## - var
##
## ****************************************
## Constraints
## ****************************************
## Leverage Constraint:
## min_sum = 1
## max_sum = 1
## actual_leverage = 1
##
## Box Constraints:
## min:
## [1] -Inf -Inf -Inf -Inf
## max:
## [1] Inf Inf Inf Inf
##
## Position Limit Constraints:
## Maximum number of non-zero weights, max_pos:
## [1] "Unconstrained"
## Realized number of non-zero weights (i.e. positions):
## [1] 4
##
## Maximum number of long positions, max_pos_long:
## [1] "Unconstrained"
## Realized number of long positions:
## [1] 1
##
## Maximum number of short positions, max_pos_short:
## [1] "Unconstrained"
## Realized number of short positions:
## [1] 3
##
##
## Diversification Target Constraint:
## [1] "Unconstrained"
##
## Realized diversification:
## [1] -114.6706
##
## Turnover Target Constraint:
## [1] "Unconstrained"
##
## Realized turnover from initial weights:
## [1] 4.430642
##
## ****************************************
## Objectives
## ****************************************
##
## Objective: return_objective
## $name
## [1] "mean"
##
## $target
## NULL
##
## $arguments
## list()
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] -1
##
## $call
## add.objective(portfolio = init.portf, type = "return", name = "mean")
##
## attr(,"class")
## [1] "return_objective" "objective"
##
## ****************************************
## Objective: portfolio_risk_objective
## $name
## [1] "var"
##
## $target
## NULL
##
## $arguments
## $arguments$portfolio_method
## [1] "single"
##
##
## $enabled
## [1] TRUE
##
## $multiplier
## [1] 1
##
## $risk_aversion
## [1] 0.25
##
## $call
## add.objective(portfolio = init.portf, type = "risk", name = "var",
## risk_aversion = 0.25)
##
## attr(,"class")
## [1] "portfolio_risk_objective" "objective"
##
## ****************************************
##
## Elapsed Time:
## Time difference of 0.009822369 secs### Visualize result
library(PerformanceAnalytics)
# Optimal weights
OptWeight = extractWeights(opt.MaxQU)
sum(OptWeight)## [1] 1chart.Weights(opt.MaxQU)
# Sharpe ratio
SharpeRatio(Return, weights = OptWeight, FUN = "StdDev")## [,1]
## StdDev Sharpe (Rf=0%, p=95%): 0.07368932# Portfolio return
PortReturn <- Return.portfolio(R = Return,
weights = OptWeight,
geometric = FALSE) #use arithmetic(FALSE) to aggregate returns
chart.TimeSeries(PortReturn)
Efficient Frontier
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear console### Load Data
library(quantmod)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"### Compute Returns
Return = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Return = na.omit(Return[-1,])
colnames(Return)<-Tickers
# Create the portfolio specification
library(PortfolioAnalytics)
init.portf <-portfolio.spec(assets = colnames(Return))
# Add a full investment constraint such that the weights sum to 1
init.portf <- add.constraint(portfolio=init.portf, type="full_investment")
# Add risk and return objective(we need them to calculate the Sharpe ratio
init.portf <- add.objective(portfolio=init.portf, type="return", name = "mean")
init.portf <- add.objective(portfolio=init.portf, type="risk", name = "StdDev")
### Efficient Frontier
MeanSd.EF <- create.EfficientFrontier(
R = Return,
portfolio = init.portf,
type = "mean-sd",
n.portfolios = 50)
summary(MeanSd.EF)## **************************************************
## PortfolioAnalytics Efficient Frontier
## **************************************************
##
## Call:
## create.EfficientFrontier(R = Return, portfolio = init.portf,
## type = "mean-sd", n.portfolios = 50)
##
## Efficient Frontier Points: 50
##
## Weights along the efficient frontier:
## IBM GE F MSFT
## 1 0.443 0.048 0.164 0.345
## 2 0.434 0.045 0.162 0.358
## 3 0.426 0.042 0.160 0.371
## 4 0.417 0.040 0.158 0.384
## 5 0.409 0.037 0.157 0.398
## 6 0.400 0.035 0.155 0.411
## 7 0.391 0.032 0.153 0.424
## 8 0.383 0.029 0.151 0.437
## 9 0.374 0.027 0.149 0.450
## 10 0.366 0.024 0.148 0.463
## 11 0.357 0.021 0.146 0.476
## 12 0.348 0.019 0.144 0.489
## 13 0.340 0.016 0.142 0.502
## 14 0.331 0.014 0.140 0.515
## 15 0.323 0.011 0.139 0.528
## 16 0.314 0.008 0.137 0.541
## 17 0.306 0.006 0.135 0.554
## 18 0.297 0.003 0.133 0.567
## 19 0.288 0.001 0.131 0.580
## 20 0.280 -0.002 0.130 0.593
## 21 0.271 -0.005 0.128 0.606
## 22 0.263 -0.007 0.126 0.619
## 23 0.254 -0.010 0.124 0.632
## 24 0.245 -0.012 0.122 0.645
## 25 0.237 -0.015 0.121 0.658
## 26 0.228 -0.018 0.119 0.671
## 27 0.220 -0.020 0.117 0.684
## 28 0.211 -0.023 0.115 0.697
## 29 0.202 -0.026 0.113 0.710
## 30 0.194 -0.028 0.112 0.723
## 31 0.185 -0.031 0.110 0.736
## 32 0.177 -0.033 0.108 0.749
## 33 0.168 -0.036 0.106 0.762
## 34 0.159 -0.039 0.104 0.775
## 35 0.151 -0.041 0.103 0.788
## 36 0.142 -0.044 0.101 0.801
## 37 0.134 -0.046 0.099 0.814
## 38 0.125 -0.049 0.097 0.827
## 39 0.116 -0.052 0.095 0.840
## 40 0.108 -0.054 0.094 0.853
## 41 0.099 -0.057 0.092 0.866
## 42 0.091 -0.060 0.090 0.879
## 43 0.082 -0.062 0.088 0.892
## 44 0.073 -0.065 0.086 0.905
## 45 0.065 -0.067 0.085 0.918
## 46 0.056 -0.070 0.083 0.931
## 47 0.048 -0.073 0.081 0.944
## 48 0.039 -0.075 0.079 0.957
## 49 0.030 -0.078 0.078 0.970
## 50 0.022 -0.080 0.076 0.983
##
## Risk and return metrics along the efficient frontier:
## mean StdDev out
## 1 0.001 0.014 0
## 2 0.001 0.014 0
## 3 0.001 0.014 0
## 4 0.001 0.014 0
## 5 0.001 0.014 0
## 6 0.001 0.014 0
## 7 0.001 0.014 0
## 8 0.001 0.014 0
## 9 0.001 0.014 0
## 10 0.001 0.014 0
## 11 0.001 0.014 0
## 12 0.001 0.014 0
## 13 0.001 0.014 0
## 14 0.001 0.014 0
## 15 0.001 0.014 0
## 16 0.001 0.014 0
## 17 0.001 0.014 0
## 18 0.001 0.014 0
## 19 0.001 0.015 0
## 20 0.001 0.015 0
## 21 0.001 0.015 0
## 22 0.001 0.015 0
## 23 0.001 0.015 0
## 24 0.001 0.015 0
## 25 0.001 0.015 0
## 26 0.001 0.015 0
## 27 0.001 0.015 0
## 28 0.001 0.015 0
## 29 0.001 0.015 0
## 30 0.001 0.015 0
## 31 0.001 0.015 0
## 32 0.001 0.015 0
## 33 0.001 0.016 0
## 34 0.001 0.016 0
## 35 0.001 0.016 0
## 36 0.001 0.016 0
## 37 0.001 0.016 0
## 38 0.001 0.016 0
## 39 0.001 0.016 0
## 40 0.001 0.016 0
## 41 0.001 0.016 0
## 42 0.001 0.016 0
## 43 0.001 0.017 0
## 44 0.001 0.017 0
## 45 0.001 0.017 0
## 46 0.001 0.017 0
## 47 0.001 0.017 0
## 48 0.001 0.017 0
## 49 0.001 0.017 0
## 50 0.001 0.017 0# Plot
chart.EfficientFrontier(MeanSd.EF,
match.col="StdDev", # which column to use for risk
RAR.text="Sharpe Ratio", # string name for risk adjusted return text
rf=0, # risk free rate
tangent.line = TRUE,
type="l", # plot line type
chart.assets=TRUE,
labels.assets=TRUE)
Fwd Looking Min Variance
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PerformanceAnalytics)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
#Calculate mean returns and covariance
MeanRet <- sapply(Returns,FUN=mean)
CovRet <- cov(Returns)
#Set expected returns 5% lower than historical avg.
ExpRet <- MeanRet*(1-.05)
#Set variances 5% higher, but covariances the same
ScaledVar <- diag(CovRet)*1.05
ExpCov <- CovRet
diag(ExpCov) <-ScaledVar
#install.packages("IntroCompFinR", repos="http://R-Forge.R-project.org")
library(IntroCompFinR)
# Minimize expected variance subject to target return(long only)
Target <- mean(MeanRet)
opt.MinVar <- efficient.portfolio(er = ExpRet,
cov.mat = ExpCov,
target.return = Target,
shorts = FALSE)
opt.MinVar## Call:
## efficient.portfolio(er = ExpRet, cov.mat = ExpCov, target.return = Target,
## shorts = FALSE)
##
## Portfolio expected return: 0.0004270109
## Portfolio standard deviation: 0.01432522
## Portfolio weights:
## IBM GE F MSFT
## 0.5049 0.0795 0.1820 0.2336barplot(opt.MinVar$weights,las=2,cex.name=.6,main="Min Variance Weights",ylim=c(0,.5))
Return.MinVar <- Return.portfolio(R=Returns, weights=opt.MinVar$weights, geometric=F)
charts.PerformanceSummary(Return.MinVar)
Fwd Looking Max Sharpe
### Housekeeping
rm(list=ls()) # clear workspace
cat("\014") # clear consolelibrary(quantmod)
library(PerformanceAnalytics)
startd = "2015-01-01"
endd = "2021-04-30"
Tickers = c("IBM","GE","F","MSFT")
getSymbols(Tickers,from=startd,to=endd,src='yahoo')## [1] "IBM" "GE" "F" "MSFT"Returns = do.call(merge,lapply(Tickers, function(x)
periodReturn(Ad(get(x)),period='daily',type='arithmetic')))
Returns = na.omit(Returns[-1,])
colnames(Returns)<-Tickers
#Calculate mean returns and covariance
MeanRet <- sapply(Returns,FUN=mean)
CovRet <- cov(Returns)
#Set expected returns 5% lower than historical avg.
ExpRet <- MeanRet*(1-.05)
#Set variances 5% higher, but covariances the same
ScaledVar <- diag(CovRet)*1.05
ExpCov <- CovRet
diag(ExpCov) <-ScaledVar
#install.packages("IntroCompFinR", repos="http://R-Forge.R-project.org")
library(IntroCompFinR)
# Maximize expected Sharpe Ratio
ExpRiskFree <- 0.75/100/30 # should match the return data frequency
opt.MaxSR <- tangency.portfolio(er = ExpRet,
cov.mat = ExpCov,
risk.free = ExpRiskFree,
shorts = FALSE)
opt.MaxSR## Call:
## tangency.portfolio(er = ExpRet, cov.mat = ExpCov, risk.free = ExpRiskFree,
## shorts = FALSE)
##
## Portfolio expected return: 0.001223585
## Portfolio standard deviation: 0.01784839
## Portfolio weights:
## IBM GE F MSFT
## 0 0 0 1barplot(opt.MaxSR$weights,las=2,cex.name=.6,main="Max Sharpe Ratio Weights",ylim=c(0,.5))
Return.MaxSR <- Return.portfolio(R=Returns, weights=opt.MaxSR$weights, geometric=F)
charts.PerformanceSummary(Return.MaxSR)